The 29th Installment“View the reverse side of a thing—Encouragement of duality”

”Dual” is a term that refers to the relationship between objects entwined with each other. The word implies, in a sense, the reverse relationship of two objects to each other or a sense of two-sidedness, and thus the dual of the dual object returns in a sense to the original object, i.e., if the dual of A is B, then the dual of B is A under a condition. The existence of the dual relationship between two objects or entities is referred to as duality. Duality is an important concept in many fields of science, engineering, and informatics.

You can see from the famous formulas of physics that duality holds true, for example, between current and voltage (potential) or between an electrical field and a magnetic field. When a proposition is expressed as a logical expression, the principle of duality can be derived from the obtained expression by replacing each AND with OR and each OR with AND, as well as each universal quantifier with an existential quantifier and each existential quantifier with a universal quantifier from the original expression, as the dual one of the original logical expression, which is used for proving propositions. As for a graph theory, in which a "graph" refers to a collection of "vertices" or "nodes" and a collection of "edges" or "arcs" that connect pairs of vertices and is widely used for expressing geographical data or network structures, a dual graph of a given planar graph, which has a vertex for each plane region of the given planar graph, and an edge for each edge in the given planar graph joining two neighboring regions, for a certain embedding of the given planar graph, may be the key for efficient graph algorithm construction.

In continuous optimization problems (linear programming and nonlinear programming problems), a duality theorem, in which a dual problem is set as the original problem (primary problem) for viewing from either of two perspectives, exists as a very indispensable theorem. In the field of discrete optimization, duality is helpful in constructing an efficient algorithm within the framework of discrete convex analysis. The more established the framework of the primal-dual algorithm, the more effective the consideration of duality for the construction of the algorithm. I think that many studies making use of duality will be pursued in the future as well.

The relationship of two-sidedness, which widely contributes to solutions to too many fields for me to state here, including science, engineering, and informatics, lurks in our immediate surroundings or in our cultural aspects.

The Japanized English phrase “a power spot” seems to be popular in Japan. The phrase refers to the special places steeped in spiritual energy (or something like that), which is helpful as a reliever in terms of feng shui or spiritualism. However, there is no scientific evidence for their effectiveness and reasons, and as a matter of fact, mere scenic and/or tourist sites seem to be lightly included in the power spots, taking advantage of the current boom. Furthermore, the number of Japanese people who visit shrines and temples on domestic tours is increasing due to the boom in history combined with that of the power spots. I do not mean to encourage clinging to matters that are not supported by scientific evidence; however, I think it may be good for people to become interested in the cultural and/or historical backgrounds even if triggered by the boom.

Ise Jingu, or Ise Grand Shrine, which is located in the city of Ise in Mie prefecture, Japan, and is familiar to Japanese as the name Oise-san from ancient times, is becoming increasingly more popular among people of various age groups other than just middle-aged and elderly persons. Not being affected by the known factor of Shikinen Sengu, the major ceremony held on a regular basis that drives the number of tourists to its peak, tourists increasingly visit Ise Jingu. Now, do you know that Ise Jingu is officially called simply Jingu and is, in fact, a shrine complex composed of 125 shrines centered on the two main shrines Kotai Jingu and Toyouke Daijingu?

The Kotai Jingu dedicated to the worship of Amaterasu-omikami, the sun goddess, is generally called Naiku (which means inside shrine in Japanese), and the Toyouke Daijingu dedicated to the worship of Toyouke-no-omikami, the goddess of food, clothing, and shelter is called Geku (which means outside shrine in Japanese). As Naiku and Geku are far apart, many tourists visit only Naiku, where there are many tourist spots nearby, and then start homeward because they are pressed for time; however, it is normally proper, as a rule from ancient times, to visit Geku first before visiting Naiku.

If you imagine the dual relationship of the two main shrines from their names, you can actually find duality in almost all aspects.

As for the deities of the two shrines, Geku relates to the ground because it is dedicated to the goddess of food, clothing, and shelter, while Naiku relates to the sun, which we can refer to as a dual relationship. The architectural style of the present chambers of both Geku and Naiku is called Yui-itsu Shinmeizukuri, which is one of the oldest architectural styles together with Taishazukuri of the Izumo Taisha Shrine. Both of the present chambers are almost the same in size. Their roofs are made of thatched reeds supported on both sides by two thick columns called the munamochi-bashira. In addition, several distinctive billets called katsuogi are located on the roof ridges at right angles. The numbers of katsuogi are nine and ten in Geku and Naiku, respectively, which shows an odd-even relationship. Odd numbers and even numbers are also called "yang number" (which means light) and "ying number" (which means shade). On both ends of the roofs, the end of the bargeboards project beyond the roofs to form the distinctive forked shape called chigi. Chigi is thought to be a relic of two interlocking timbers that were left uncut when the roof was built in ancient times. In Geku, the end of the chigi is cut perpendicular to the ground (it is called "soto-sogi", which means sharpened outside), while in Naiku, the end of the chigi is cut horizontally (called "uchi-sogi", which means sharpened inside), where you can find the relationship of duality. Apart from the two main chambers, fine figures of duality is found in the other shrines, such as the odd-even relationship in the number of katsuogi and the rule in the vertical or horizontal cutting of chigi.

The duality, or two-sidedness valued by ancient people—I think it is important to view everything from both sides. When you find difficulty, I hope you will reverse it without sticking to the surface or looking at it from a biased point of view. The structure found in the backside may give you some tips for a solution.